 
Summary: From Grushin to Heisenberg via an
isoperimetric problem
Nicola Arcozzi and Annalisa Baldi
Abstract
The Grushin plane is a right quotient of the Heisenberg group.
Heisenberg geodesics' projections are solutions of an isoperimetric
problem in the Grushin plane.
1 Introduction
It is a known fact that there is a correspondence between isoperi
metric problems in Riemannian surfaces and subRiemannian geome
tries in threedimensional manifolds. The most significant example
is the isoperimetric problem in the plane, corresponding to the sub
Riemannian geometry of the Heisenberg group H.
We briefly recall this connection following the exposition in [Mont].
Consider, on the Euclidean plane, the oneform = 1
2 (xdy  ydx),
which satisfies d = dx dy and which vanishes on straight lines
through the origin. By Stokes' Theorem, the signed area enclosed by
a curve is . Let c : [a, b] R2 be a curve. For each s in [a, b],
let s be the union of the curve c restricted to [a, s], of the segment
